Question

Suppose that we observe the following spot rates, i.e. the yield curve is upward sloping. The spot rates are annual rates that are semi-annually compounded.

Time to Maturity	Spot Rate
0.5	2.00%
1.0	2.50%
1.5	3.00%
2.0	3.50%

1.  Compute the six-month forward curve, i.e. compute f(0,0.5,1.0), f(0,1.0,1.5), f(0,1.5,2.0).
2.  What can we say about the forward curve?
When the term structure of interest rates is upward sloping, the forward curve is __________ (upward/downward) sloping.

Answer 1

We have to compute

f(0,0.5,1.0) i.e the 6 month forward rate 6 months from now

f(0,1.0,1.5) i.e the 6 month forward rate 1 year from now

f(0,1.5,2.0). i.e the 6 month forward rate 1 .5 years from now

Now,

1) f(0,0.5,1.0) =

  $\frac{(1+0.025/2)^2}{1+0.02/2} - 1$   (Note: we have to convert the annualised spot rates to semi annual rates)

= 0.0150 or 1.5%

Now this is the semiannual rate,so the required forward rate = 2* 1.5 = 3%

2) f(0,1.0,1.5) =

  

(1+0.03/2} (1+0.025/22 1

    = 0.0200 or 2%

Now this is the semiannual rate,so the required forward rate = 2* 2 = 4%

f(0,1.5,2.0) =

$\frac{(1+0.035/2)^4}{(1+0.03/2)^3}$

= 0.0250 or 2.5%

Now this is the semiannual rate,so the required forward rate = 2* 2.5 = 5%

When the term structure of interest rates is upward sloping, the forward curve is upward sloping as is seen in this problem.